We also make use of some properties from linear algebra.
1) For any matrix \(H\) , it holds that \(H^\dagger = H^\top (HH^\top )^\dagger \)  [1]}. 2) For a matrix \(A \in \mathbb {R}^{n\times m}\) , and \(B K A\) , where \(K\in \mathbb {R}^{n\times n}\)  is an invertible matrix, it holds that \(A^\dagger A = B^\dagger B\) . 3) Let \(A, B \in \mathbb {R}^{n\times m}\)  such that \(\textup {range}(A^\top ) = \textup {null}(B)\) . Then it holds that \(A^\dagger A + B^\dagger B = I_m\) . 4) For matrices \(A,B\in \mathbb {R}^{n\times m}\) , \(A\)  is left equivalent (or row equivalent) to \(B\)  if and only if there exists an invertible matrix \(P\in \mathbb {R}^{n\times n}\)  such that \(A= PB\) . It then can be shown that \(A\)  and \(B\)  are left-equivalent if and only if \(\textup {null}(A) = \textup {null}(B)\) .
